Problem 70 - Algebra for College Students

Question

Suppose we want to find the coordinates of the midpoint of a line segment. Let \(P(x, y)\) represent the midpoint of the line segment from \(A\left(x_{1}, y_{1}\right)\) to \(B\left(x_{2}, y_{2}\right)\). Using the method in Problem 68, the formula for the \(x\) coordinate of the midpoint is \(x=x_{1}+\frac{1}{2}\left(x_{2}-x_{1}\right)\). This formula can be simplified algebraically to produce a simpler formula. $$ \begin{aligned} &x=x_{1}+\frac{1}{2}\left(x_{2}-x_{1}\right) \\ &x=x_{1}+\frac{1}{2} x_{2}-\frac{1}{2} x_{1} \\ &x=\frac{1}{2} x_{1}+\frac{1}{2} x_{2} \\ &x=\frac{x_{1}+x_{2}}{2} \end{aligned} $$ Hence the \(x\) coordinate of the midpoint can be interpreted as the average of the \(x\) coordinates of the endpoints of the line segment. A similar argument for the \(y\) coordinate of the midpoint gives the following formula. $$ y=\frac{y_{1}+y_{2}}{2} $$ For each of the pairs of points, use the formula to find the midpoint of the line segment between the points. (a) \((3,1)\) and \((7,5)\) (b) \((-2,8)\) and \((6,4)\) (c) \((-3,2)\) and \((5,8)\) (d) \((4,10)\) and \((9,25)\) (e) \((-4,-1)\) and \((-10,5)\) (f) \((5,8)\) and \((-1,7)\)

Step-by-Step Solution

Answer

Midpoints are (5, 3), (2, 6), (1, 5), (6.5, 17.5), (-7, 2), and (2, 7.5).

Step 1: Understand the Midpoint Formula

The formula to find the midpoint \( P(x, y) \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is: \( x = \frac{x_1 + x_2}{2} \) and \( y = \frac{y_1 + y_2}{2} \). This formula calculates the average of the \( x \) and \( y \) coordinates of the endpoints.

Step 2: Apply the Formula to Part (a)

For the points \( (3, 1) \) and \( (7, 5) \):Calculate the \( x \) coordinate: \[ x = \frac{3 + 7}{2} = \frac{10}{2} = 5 \]Calculate the \( y \) coordinate: \[ y = \frac{1 + 5}{2} = \frac{6}{2} = 3 \]Thus, the midpoint is \( (5, 3) \).

Step 3: Apply the Formula to Part (b)

For the points \( (-2, 8) \) and \( (6, 4) \):Calculate the \( x \) coordinate: \[ x = \frac{-2 + 6}{2} = \frac{4}{2} = 2 \]Calculate the \( y \) coordinate: \[ y = \frac{8 + 4}{2} = \frac{12}{2} = 6 \]Thus, the midpoint is \( (2, 6) \).

Step 4: Apply the Formula to Part (c)

For the points \( (-3, 2) \) and \( (5, 8) \):Calculate the \( x \) coordinate: \[ x = \frac{-3 + 5}{2} = \frac{2}{2} = 1 \]Calculate the \( y \) coordinate: \[ y = \frac{2 + 8}{2} = \frac{10}{2} = 5 \]Thus, the midpoint is \( (1, 5) \).

Step 5: Apply the Formula to Part (d)

For the points \( (4, 10) \) and \( (9, 25) \):Calculate the \( x \) coordinate: \[ x = \frac{4 + 9}{2} = \frac{13}{2} = 6.5 \]Calculate the \( y \) coordinate: \[ y = \frac{10 + 25}{2} = \frac{35}{2} = 17.5 \]Thus, the midpoint is \( (6.5, 17.5) \).

Step 6: Apply the Formula to Part (e)

For the points \( (-4, -1) \) and \( (-10, 5) \):Calculate the \( x \) coordinate: \[ x = \frac{-4 + (-10)}{2} = \frac{-14}{2} = -7 \]Calculate the \( y \) coordinate: \[ y = \frac{-1 + 5}{2} = \frac{4}{2} = 2 \]Thus, the midpoint is \( (-7, 2) \).

Step 7: Apply the Formula to Part (f)

For the points \( (5, 8) \) and \( (-1, 7) \):Calculate the \( x \) coordinate: \[ x = \frac{5 + (-1)}{2} = \frac{4}{2} = 2 \]Calculate the \( y \) coordinate: \[ y = \frac{8 + 7}{2} = \frac{15}{2} = 7.5 \]Thus, the midpoint is \( (2, 7.5) \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebra to describe geometric principles. It allows us to use a coordinate system, typically the Cartesian coordinate system, to represent geometric figures as equations.
This system is an invaluable tool for calculating the relationships between points, lines, and surfaces. With coordinate geometry, tasks such as finding distances, angles, and midpoints between points become more manageable, as they are translated into solvable algebraic problems.
In the context of finding a midpoint, coordinate geometry simplifies the process by allowing us to represent endpoints of a line segment with coordinates, which can then be used to calculate an average. This transformation from a geometric problem into an algebraic one highlights the versatility and power of coordinate geometry.

Line Segments

A line segment is a fundamental concept in geometry, defined as the part of a line that is bounded by two distinct end points. These endpoints can be used to calculate various properties of the line segment, such as its length and midpoint.
Understanding line segments and their properties is crucial because they form the building blocks for more complex shapes and figures in geometry. Midpoints, in particular, are of interest when dealing with line segments, as they help to find the center location between the two endpoints.

• To find the midpoint, use the coordinates of the endpoints.
• Apply the midpoint formula on the coordinates.

The midpoint itself lies equidistant from both endpoints, effectively dividing the segment into two equal parts. This concept forms the basis of many applications in mathematics and related fields.

Algebraic Simplification

Algebraic simplification refers to the process of reducing a complex algebraic expression into its simplest or most efficient form. This is achieved by applying rules and properties of algebra to combine like terms, factor expressions, and utilize identities.
For instance, consider the expression for the midpoint's x-coordinate given by \[ x = x_{1} + \frac{1}{2}(x_{2} - x_{1}) \].
By expanding this and rearranging terms, we simplify to the more intuitive form \[ x = \frac{x_{1} + x_{2}}{2} \].
The simplified formula represents the average of the x-coordinates of the endpoints, which is much easier to interpret and apply.

• Simplification helps prevent errors.
• Makes formulas easier to remember and use.

Simplifying expressions is not only a useful skill but also a fundamental step in solving many algebraic and geometric problems.

Average of Numbers

The average of numbers is a mathematical concept commonly used to find the central value in a set of numbers. In the context of geometry, specifically when calculating a midpoint, the average provides the central ordinate values that define the midpoint.
The formula \[ x = \frac{x_{1} + x_{2}}{2} \] for the x-coordinate and \[ y = \frac{y_{1} + y_{2}}{2} \] for the y-coordinate of the midpoint reflects this principle. These formulas calculate the average of the x and y coordinates of the endpoints, respectively.

• Use sum of values in the set.
• Divide by the number of values (2 for midpoints).

Averaging is pivotal in ensuring that the midpoint is indeed centered between the two endpoints, thus serving its function as the geometric center of the line segment. Understanding how averages work is fundamental in grasping how midpoints are determined, giving insight into the balance between positions along a line.